Nonparametric empirical Bayes estimation of the matrix parameter of the Wishart distribution

We consider independent pairs (X-1, Sigma(1)), (X-2, Sigma(2)), ..., (X-n, Sigma(n)), where each Sigma(i) is distributed according to some unknown density function g(Sigma) and, given Sigma(i) = Sigma, X-i has conditional density function q(xΣ) of the Wishart type. In each pair the first component is observable but the second is not. After the (n+1)th observation Xn+1 is obtained, the objective is to estimate Sigma(n+1) corresponding to Xn+1. This estimator is called the empirical Bayes (EB) estimator of Sigma. An EB estimator of Sigma is constructed without any parametric assumptions on g(Sigma). Its posterior mean square risk is examined, and the estimator is demonstrated to be pointwise asymptotically optimal.